Biology Reference
In-Depth Information
binomial distribution with either constant k or, perhaps more realistically,
k linearly dependent on the modeled mean.
57
The latter approach of deriving distributional properties from the
underlying modeled population processes is more akin to Cassie's idea of
the fundamental (or mechanistic) processes that underlie observed
epidemiological patterns, and requires either that the dynamic model has
been formulated stochastically at an individual level or that it is possible
to infer or derive distributional properties from the deterministic
formulation. For example, a frequently applied interpretation of the
deterministic immigration
death process is the following.
56,64,111,184,185
The population rate of parasite establishment,
e
, is considered as the
average of individual rates of homogeneous or inhomogeneous Poisson
processes; an individual rate,
L
L
i
, is given by the population (average) rate
multiplied by a “susceptibility” factor,
; susceptibility factors are
assumed to be gamma distributed with mean 1 and variance 1/k; the
resulting marginal distribution of worms among individuals is negative
binomial with mean
L
i
¼
s
i
L
L
and overdispersion parameter k. Thus,
the
example of
death model fitted to individual
A. lumbricoides age-worm burden data with constant overdispersion
parameter k has plausible mechanistic (or fundamental) interpretation.
Generally, it is not possible to derive such neat analytical results from
either explicitly modeled or inferred underlying stochastic processes.
Indeed the marginal distribution will seldom conform to a closed-form
probability distribution. This is particularly the case with longitudinal
data where one has a joint distribution of repeated correlated observa-
tions made on the same individual and independent observations made
from different individuals. In such circumstances fitting becomes a chal-
lenging task, reliant on computational-intensive methods. One such
approach is to formulate a dynamic model in discrete time, modeling
explicitly the stochastic evolution of the model using a state-space
construction. Combining model predictions
an immigration
e
e
conditional on current
parameter values
with observed data at specific times (or host ages)
permits evaluation of the likelihood. Consequently, Markov chain Monte
Carlo methods can be used to evaluate the posterior distribution of the
unknown parameter. For an introduction to state-space models see
Chapter 11 of Bolker
186
and references therein.
e
Dynamic Models of Infection and Transmission
The development of mathematical, dynamic models for helminth
infections was led by the pioneering contributions of Anderson and
colleagues,
64,119,120,187,188
beginning in earnest in the 1970s and 1980s to
understand theoretical and applied questions on helminth transmission
dynamics and population biology (Chapter 9). Since this time, much
